Numerical Modeling of Silo Filling.?I: Continuum Analyses

Two main computational methods have been used in recent years to model the behavior of particulate solids in silos. These are the finite element method and the discrete element method. To assess the current state of the art in the two methods applied to silo problems, and to evaluate their capabilities without bias, an international collaborative project was set up to compare predictions of several silo phenomena. The first of these computational challenges was deemed the simplest: that of filling a silo or container with particulate solid. This paper presents an overview of the findings of this first problem, based on a total of 38 independent calculations. Here the collaborative project is briefly outlined and some deduced outcomes from calculations by continuum analysts are compared. The results of discrete element calculations are described in a companion paper, which also compares the two methods and comments on their strengths and weaknesses.     

Partitioned Distinct Element Method Simulation of Granular Flow within Industrial Silos

A method for performing discrete element simulations of granular flow and pressures within industrial silos is presented. Special attention is devoted to complex problems involving large numbers of particles and sophisticated boundary conditions due to the presence of inserts. The proposed method consists of partitioning the silo into layers that are analyzed sequentially, and in determining stresses and velocities at the virtual interlayer boundaries. The method is first validated by simulating the discharge of a single insert hopper containing 20,000 particles, performing both a simulation of the whole silo and a multilayer partition. The results show a small discrepancy in the displacement fields produced by the two simulations. Then the discharge of an industrial silo containing 170,000 particles with several inserts of different size and shape is simulated. The relevance of the stress and velocity fields obtained confirms the feasibility and the efficiency of the procedure. The method allows for managing huge numbers of particles with a limited memory capacity and a gain of computational time that may be significant depending on each particular case.     

颗粒流数值模拟的现状

介绍了颗粒物质与颗粒流的概念以及目前颗粒流数值模拟的现状;按照采用的模型是否连续,将模型分为:连续介质模型和非连续介质模型两类;着重介绍了非连续介质模型中的离散元法(DEM):硬球模型和软球模型.通过颗粒流的数值模拟,可以对料仓、固定床、移动床和流化床等设备的设计和操作进行优化,因此在工程中具有重要意义.     

Discharge and the Eccentricity of the Hopper Influence on the Silo Wall Pressures

The objective of this work is to perform an analysis of pressure distributions in grain silos for both the discharge and static conditions with eccentric outlets. To this end, the commercial ANSYS program, based on the finite-element method is employed. In this paper, results are presented for a silo with a height of 10.5 m (cylinder height 8 m, base cone height 2.5 m), a silo radius of 3 m, and an outlet radius of 0.5 m. Outlet eccentricity ranged from 0% (central outlet) to 100% (maximally eccentric outlet, tangential to the silo cylinder). Based on the results, this paper proposes new methods of analyzing silo discharge and the influence of outlet eccentricity. The discharge model combines both dilatancy and type of discharge effects. 3D models were developed to analyze the eccentricity of the outlet. The results are compared with previous research data and with standard design guidelines.     

立式砂仓尾砂体积分数随砂仓高度变化规律研究

针对现有放砂系统存在的砂浓度波动大、实际放砂浓度较低且不稳定等问题,提出了一种新的有别于立式砂仓传统放砂工作模式的放砂模型,即通过控制给料流量和浓度、放砂流量及尾砂堆积面高度3个条件达到给料、溢流、放砂同时进行的连续放砂模型,之后运用流体力学软件对连续放砂模型进行模拟分析。结果表明：当物料达到进出平衡时,不同尾砂堆积面高度对应的底流放砂浓度不同;砂浆中尾砂体积分数随砂仓高度呈规律性变化,由此提出预测模型。在云南玉溪大红山铜矿进行了连续放砂工业试验,结果表明工业试验效果良好,证实了连续放砂的可行性。     

Modeling Nonspherical Particles Using Multisphere Discrete Elements

In this paper axisymmetrical particles are modeled as multisphere discrete elements using a method in which particles are represented by overlapping spheres, fixed rigidly with respect to a local coordinate system. Contact detection is sphere-based and the resultant forces are transformed to the particle centroid to calculate the particle motion using standard discrete element method conventions. The multisphere method was used to model discharge of ellipse-shaped particles through an orifice in a flat-bottomed hopper and the simulations compared with physical experiments at the same scale. There was good agreement between the flow behavior of the simulated and physical particle assemblies for all orifice sizes. The rate of discharge and the vertical velocity profiles in the region of converging flow determined for the simulated and physical flows were in close agreement for flow from the larger orifices. A similar relationship between frequency of arch formation, particle mean diameter, and orifice diameter as for spheres was observed.     

Numerical Aspects of FE Simulations of Granular Flow in Silos

Some important numerical aspects concerning the finite-element simulation of granular flow are presented. The main topic of this paper is the so-called “switch” phenomenon, which is a stress peak at the junction of the vertical and the inclined part of a mass flow silo during its discharge. Case studies prove that the FE simulation of silos and simplified systems should be handled with great care, as many wall pressure peaks occurring at places where the shape of the silo is changing may well be caused by numerical corner singularities. Other numerical aspects lead to consequences for the filling process and the opening phase of silo emptying.     

Locating Discharge Trajectories in Still and Moving Ambient Fluids

The angular momentum principle is employed to locate the trajectories of wastewater plumes. This momentum-based method differs from the traditional approach where a perturbation analysis, based on the centerline velocities, is employed for locating discharges. Evidence of the latter can be found in existing Eulerian-integral and length-scale models. The momentum-based method is incorporated into the hybrid model SD3D, where the regional flow solutions are modified to incorporate the influences of relatively small components of momentum on the discharge trajectory. This method provides a clear understanding of the factors that influence the location of the discharge. The momentum-based approach yields analytical trajectory solutions in many cases, and it eliminates the need to arbitrarily select the appropriate characteristic velocity for locating the flow. Comparisons are made with available experimental data, and they show that the momentum-based method provides accurate predictions of the flow trajectories under a variety of discharge conditions. Comparisons are also made with predictions from the CORMIX1 and JETLAG models. In general the predictions are consistent, but some important discrepancies are highlighted. The use of the momentum-based method for locating discharges is discussed in light of recent experimental studies.     

Finite-Element Modeling of Filling Pressures in a Full-Scale Silo

A detailed nonlinear finite-element analysis is undertaken of the wall pressures exerted by iron ore pellets in a full-scale silo experiment. The ring stiffeners on the exterior of the silo wall cause local small axisymmetric geometric imperfections, and the effect of these on the pressure predictions is explored, together with its sensitivity to the chosen bulk solid constitutive model. This paper focuses on the filling pressures alone, because these are a necessary starting point if discharge pressure predictions are to be made with confidence.     

3D Finite-Element Simulation of a Square Silo with Flexible Walls

Most of the studies that have been published to evaluate stresses on silo walls during filling or discharge stages are based on a rigid wall assumption. In a 2D approach, the wall flexibility can be approximately modeled by using a corrective factor applied to the whole pressure distribution. It has been shown that such models are in agreement with experimental measurements on circular silos. But in square or rectangular silos, the variation of the stiffness of the wall due to vertical or horizontal stiffeners produces nonuniform wall deformations that are not taken into account in the previous axisymmetric 2D model. The aim of this paper is to present a full 3D modeling of the filling and discharge stages using a nonlinear finite-element method. The bulk material behavior is based on an elastoplastic law. Contact elements using a Mohr-Coulomb criterion simulate the interaction between the wall and the bulk material, and the flexibility of the silo structure is modeled with beam and shell elements. A detailed analysis of the numerical results computed for a square silo filled with wheat and discharged through a central outlet is presented and discussed.     

Flow Pattern for a Silo with Two Layers of Materials with Single or Double Openings

For the silo flow of two materials with contrasting strengths, an automatic separation without mixing of the materials is observed during drawing. In the present paper, the writers observe that a columniform funnel flow will appear in a silo with a single opening while a uniform flow will occur for a silo with a double opening. The classical and popular Janssen equation is hence less suitable for a silo with a single opening than for a silo with a double opening. The development of the arch action arising from the contact force is investigated through the distinct element analysis for the two different types of silo. It is also found that there are significant differences in the velocity and stress fields between the two different types of silo, which are worth consideration in the design of the silo and the recovery of the mineral ore.     

Modeling the Transient Pumping from Two Aquifers Using MODFLOW

A procedure is proposed for calculating the spatial and temporal variation of drawdown due to pumping a well tapping two aquifers separated by an aquitard, using convolution and MODFLOW. It can take into account the unsteady pumping discharge and cross flow through the intervening aquitard. A discrete pulse kernel method based on superposition/convolution is used to account for the unsteady pumping discharge. The discrete pulse kernels are calculated using MODFLOW. The contributions of the aquifers to the pumped discharge are accounted implicitly and not required to be specified explicitly. Available numerical models (e.g., MODFLOW) require the aquifer contributions that are implicitly controlled, to be specified explicitly. The use of the suggested procedure is illustrated using examples. The contributions of the aquifers are found not in proportion to their transmissivities but vary with time, when the diffusivities of the aquifers are not equal. Applying the new procedure, the numerical models, such as MODFLOW can be used to correctly model the transient pumping from two aquifers with cross flow; thus, it opens up the possibility of numerically accounting for the aquifer heterogeneity while dealing with the flow to a well tapping two aquifers under a transient pumping, which would be otherwise difficult to account for analytically.     

Flow Curve and Failure Modeling for High‐Mn Steels on a Microstructural Scale

A microstructural model for flow curve and failure modeling based on the representative volume element (RVE) approach is developed for high‐Mn steels. The polycrystalline structure is generated by discrete Voronoi tessellation. Physically based material models for mechanical twinning and the ε‐martensite phase transformation are implemented for describing the hardening behavior. A ductile damage model based on the multiaxial state of strain calculates the fracture and failure. Uniaxial tensile tests were carried out for the steel 22Mn0.6C with varying grain size and for a temperature range of 123–423 K. Experimental results are in good agreement with the calculated RVE models.     

Nonlinear PLS Method for Side Weir Flows

Side weirs are flow-regulating devices commonly encountered in hydraulic engineering. In the past, the discharge coefficient for flow past a side weir was investigated experimentally by many researchers. In this study, a modified discharge coefficient Cd for side weirs in rectangular channels and circular channels is defined. The multivariable nonlinear partial least square (PLS) method is used to determine the empirical equations relating Cd with the dimensionless weir parameters F1, S/Y1, and L/D. Compared to the previous studies, the procedures to calculate the discharge of the side weirs is simplified. The discharge predicted using the empirical equations based on the nonlinear PLS method is in good agreement with the measured discharge. The nonlinear PLS method can also be applied to many other hydraulic flow configurations characterized by a large number of variables.     

Critical Flow over Circular Crested Weirs

The critical flow principle is a useful approach for the hydraulic analysis of round-crested weirs due to their single head-discharge relationships. The hydraulics of circular-crested weirs is examined using simplified models incorporating streamline curvature effects, comparing their predictions with experimental data. A generalized one-dimensional model based on the critical flow in curvilinear motion has been developed. The discharge coefficient increases with the specific energy normalized with the radius of curvature, E/R, when streamline curvature effects are included. The relative flow depth at the crest decreases as E/R increases. The flow at the weir crest is only critical for a normalized specific energy value of E/R ≈ 0.5–0.6. For larger heads, the flow at the weir crest has been found to be supercritical.     

Influence of Hopper Eccentricity on Discharge of Cylindrical Mass Flow Silos with Rigid Walls

The influence of hopper eccentricity on silo discharge has always concerned researchers. Eccentric discharge brings about asymmetries in the distribution of normal pressures, the values of which are not known with precision. This study presents a three-dimensional model of the dynamic discharge of a silo with an eccentric hopper, considering the Drucker-Prager plasticity model for the behavior of stored material. The results obtained for different hopper eccentricities reveal that pressures decrease in the hopper along the generatrix of the silo above the outlet, as compared to the opposite side. An increase in normal pressures occurs at this generatrix, as compared to the diametrically opposite generatrix, in the lower part of the cylindrical wall above the silo-hopper transition.     

Flow Variation of Duckbill Valve Jet in Relation to Large Elastic Deformation

Duckbill-shaped elastomer valves are often installed on wastewater effluent diffusers, stormwater outfalls, and industrial flow systems to prevent backflow and sediment/salt water intrusion. Unlike fixed diameter nozzles, the flow from a duckbill valve (DBV) depends both on the driving pressure and the size of the valve opening. A nonlinear large deformation finite element analysis of a prototype DBV is reported herein. The elastomer is modeled as a hyperelastic incompressible solid, and the flow inside the DBV, shaped like a converging nozzle, is treated as energy conserving. The deformed valve is computed iteratively from sequential standard large deformation analysis of the internal flow and pressure loading. The calculations show that the valve opening is lip shaped, and the maximum stress occurs around the two sides of the saddle of the DBV; maximum strains are on the order of 5%. In contrast to the traditional square-root head–discharge dependence, a linear pressure–discharge relation is predicted for a range of elastomer thickness; the jet velocity/valve opening area varies nonlinearly with discharge. The normalized predictions of valve discharge flow and opening area as a function of the driving pressure are in excellent agreement with experimental data.     

Wall Shear Stress in the Early Stage of Unsteady Turbulent Pipe Flow

Conventionally, wall shear stress in an unsteady turbulent pipe flow is decomposed into a quasi-steady component and an “unsteady wall shear stress” component. Whereas the former is evaluated by using “standard” steady flow correlations, extensive research has been carried out to develop methods to predict the latter leading to various unsteady friction models. A different approach of decomposition is used in the present paper whereby the wall shear in an unsteady flow is split into the initial steady value and perturbations from it. It is shown that in the early stages of an unsteady turbulent pipe flow, these perturbations are well described by a laminar-flow formulation. This allows simple expressions to be derived for unsteady friction predictions, which are in good agreement with experimental and computational results.     

Empirically Based Gamma-Distributed Random Wall Pressure Field in Silo

Measurements show that silo wall pressures exhibit large fluctuations in time and space during discharge of the silo. This observation is important for the design of the silo wall because spatial pressure variations may impose substantial bending moments in the silo wall that otherwise may be small or vanishing due to the carrying ability of the membrane forces in the silo wall. Information about the stochastic properties of this pressure variation cannot be obtained from any existing continuum model for the silo medium flowing within the confinement of the silo. Therefore the modeling must presently be tied to statistical analyses of the empirical evidence combined with simple mechanical principles. This paper analyzes the same data set as that analyzed by Munch-Andersen et al., but on the basis of a completely different and physically less disputable stochastic pressure field model than the one considered by them. It is shown that an explicitly constructed gamma distribution type of field in equilibrium with itself fits well to the measurements made in the Swedish Karpalund silo.